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Regression with random design: A minimax study

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  • Chesneau, Christophe

Abstract

The problem of estimating a regression function based on a regression model with (known) random design is considered. By adopting the framework of wavelet analysis, we establish the asymptotic minimax rate of convergence under the risk over Besov balls. A part of this paper is devoted to the case where the design density is vanishing.

Suggested Citation

  • Chesneau, Christophe, 2007. "Regression with random design: A minimax study," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 40-53, January.
    • Handle: RePEc:eee:stapro:v:77:y:2007:i:1:p:40-53
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    References listed on IDEAS

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    1. Gérard Kerkyacharian & Dominique Picard & Lucien Birgé & Peter Hall & Oleg Lepski & Enno Mammen & Alexandre Tsybakov & G. Kerkyacharian & D. Picard, 2000. "Thresholding algorithms, maxisets and well-concentrated bases," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 283-344, December.
    2. Cai, T. Tony & Brown, Lawrence D., 1999. "Wavelet estimation for samples with random uniform design," Statistics & Probability Letters, Elsevier, vol. 42(3), pages 313-321, April.
    3. Iain M. Johnstone & Gérard Kerkyacharian & Dominique Picard & Marc Raimondo, 2004. "Wavelet deconvolution in a periodic setting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 547-573, August.
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    Cited by:

    1. Christophe Chesneau, 2014. "A Note on Wavelet Estimation of the Derivatives of a Regression Function in a Random Design Setting," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-8, April.
    2. Kou, Junke & Liu, Youming, 2016. "An extension of Chesneau’s theorem," Statistics & Probability Letters, Elsevier, vol. 108(C), pages 23-32.

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